Gorenstein-Projective Modules over Upper Triangular Matrix Artin Algebras

Abstract

Gorenstein-projective module is an important research topic in relative homological algebra, representation theory of algebras, triangulated categories, and algebraic geometry (especially in singularity theory). For a given algebra $A$ , how to construct all the Gorenstein-projective $A$ -modules is a fundamental problem in Gorenstein homological algebra. In this paper, we describe all complete projective resolutions over an upper triangular Artin algebra $\mathrm{\Lambda }=\left(\begin{array}{cc}A& {}_A{{\displaystyle M}}_B\\ 0& B\end{array}\right)$ . We also give a necessary and sufficient condition for all finitely generated Gorenstein-projective modules over $\mathrm{\Lambda }=\left(\begin{array}{cc}A& {}_A{{\displaystyle M}}_B\\ 0& B\end{array}\right)$ .